平衡二叉树（Self-balancing Binary Search Tree）

Date: 2019-04-11 18:49:18

AVL树的基本操作

 //存储结构
 struct node
 {
     int data;
     int height; //记录当前子树的高度(叶子->根)
                 //存储平衡因子的话，无法通过其子树算得该树的平衡因子
     node *lchild, *rchild;
 };

 //新建结点
 node *newNode(int v)
 {
     node *root = new node;
     root->data = v;
     root->height = ;
     root->lchild = root->rchild = NULL;
     return root;
 }

 //获取当前结点所在高度
 int GetHeight(node *root)
 {
     if(root == NULL)
         return ;
     return root->height;
 }

 //计算结点的平衡因子
 int GetBalanceFactors(node *root)
 {
     return GetHeight(root->lchild)-GetHeight(root->rchild);
 }

 //更新结点高度
 void UpdataHeight(node *root)
 {
     root->height = max(GetHeight(root->lchild), GetHeight(root->rchild))+;
 }

 //查找
 void Search(node *root, int x)
 {
     if(root == NULL)
         return;
     if(x == root->data)
         //visit
     else if(x < root->data)
         Search(root->lchild, x);
     else
         Search(root->rchild, x);
 }

 //左右旋互为逆操作
 //左旋
 void LeftRotation(node *&root)
 {
     node *temp = root->lchild;      //temp指向新的根结点B
     root->rchild = temp->lchild;    //B的左子树给A的右子树
     temp->lchild = root;            //B的左子树变为A
     UpdataHeight(root);             //更新结点高度
     UpdataHeight(temp);
     root = temp;                    //令B成为新的根结点
 }

 //右旋
 void RightRotation(node *&root)
 {
     node *temp = root->lchild;
     root->lchild = temp->rchild;
     temp->rchild = root;
     UpdataHeight(root);
     UpdataHeight(temp);
     root = temp;
 }

 /*
 1.LL: A==+2, A->lchild=+1
     A作为root进行右旋
 2.LR: A==+2, A->lchild=-1
     A->lchild作为root进行左旋 --> 转化为LL
     A作为root进行右旋
 3.RR: A==-2, A->rchild=-1
     A作为root进行左旋
 4.RL: A==-2, A->rchild=+1
     A->rchild作为root进行右旋 --> 转化为RR
     A作为root进行左旋
 */

 //插入
 void Insert(node *&root, int v)
 {
     if(root == NULL)
     {
         root = newNode(v);
         return;
     }
     if(v < root->data)
     {
         Insert(root->lchild, v);
         UpdataHeight(root);     //更新树高
         if(GetBalanceFactor(root) == )
         {
             if(GetBalanceFactor(root->lchild) == )
                 RightRotation(root);
             else
             {
                 LeftRotation(root->lchild);
                 RightRotation(root);
             }
         }
     }
     else
     {
         Insert(root->rchild, v);
         UpdataHeight(root);
         if(GetBalanceFactor(root) == -)
         {
             if(GetBalanceFactor(root->rchild) == -)
                 LeftRotation(root);
             else
             {
                 RightRotation(root->rchild);
                 LeftRotation(root);
             }
         }
     }
 }

 //建立
 node *Create(int data[], int n)
 {
     node *root = NULL;
     for(int i=; i<n; i++)
         Insert(root, data[i]);
     return root;
 }

判断一棵树是否为AVL树

 #include <cstdio>
 const int M = ;
 int pre[M]={,,,,,,,,,};
 int in[M]={,,,,,,,,,};
 struct node
 {
     int data;
     node *lchild, *rchild;
 };

 node *Create(int preL, int preR, int inL, int inR)
 {
     if(preL > preR)
         return NULL;
     node *root = new node;
     root->data = pre[preL];
     int k;
     for(k=inL; k<=inR; k++)
         if(in[k] == root->data)
             break;
     int numLeft = k-inL;
     root->lchild = Create(preL+, preL+numLeft, inL, k-);
     root->rchild = Create(preL+numLeft+, preR, k+, inR);
 }

 int IsAvl = true;
 int IsAVL(node *root)
 {
     if(root == NULL)
         return -;
     int bl = IsAVL(root->lchild)+;
     int br = IsAVL(root->rchild)+;
     if(bl-br> || bl-br<-)
         IsAvl = false;
     return bl>br?bl:br;
 }

 int main()
 {
 #ifdef    ONLINE_JUDGE
 #else
     freopen("Test.txt", "r", stdin);
 #endif

     node *root = Create(,M-,,M-);
     IsAVL(root);
     if(IsAvl)
         printf("Yes.");
     else
         printf("No.");

     return ;
 }